Galois orbit of Artin representations 4.2e6_3e2_5e2_11e2.8t22.8

• Label: 4.2e6_3e2_5e2_11e2.8t22.8
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1742400= 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 9 x^{6} - 8 x^{5} + 31 x^{4} + 36 x^{3} - 23 x^{2} - 28 x + 16 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 4 + 15\cdot 31 + 29\cdot 31^{2} + 9\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 9 + 26\cdot 31 + 26\cdot 31^{2} + 11\cdot 31^{3} + 12\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 10 + 2\cdot 31 + 7\cdot 31^{2} + 24\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 14 + 5\cdot 31 + 5\cdot 31^{2} + 15\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 16 + 29\cdot 31 + 13\cdot 31^{2} + 28\cdot 31^{3} +O\left(31^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 20 + 22\cdot 31 + 23\cdot 31^{2} + 11\cdot 31^{3} + 23\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 22 + 24\cdot 31 + 4\cdot 31^{2} + 25\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 29 + 28\cdot 31 + 12\cdot 31^{2} + 28\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(5,7)(6,8)$

$(3,4)(6,8)$

$(1,2)(6,8)$

$(1,6)(2,8)(3,5)(4,7)$

$(1,8)(2,6)(3,5)(4,7)$

$(1,5)(2,7)(3,6)(4,8)$

$(1,5)(2,7)(3,8)(4,6)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,2)(3,4)(5,7)(6,8)$	$-4$
$2$	$2$	$(1,2)(6,8)$	$0$
$2$	$2$	$(1,6)(2,8)(3,5)(4,7)$	$0$
$2$	$2$	$(1,5)(2,7)(3,6)(4,8)$	$0$
$2$	$2$	$(1,8)(2,6)(3,5)(4,7)$	$0$
$2$	$2$	$(1,3)(2,4)(5,6)(7,8)$	$0$
$2$	$2$	$(3,4)(6,8)$	$0$
$2$	$2$	$(1,2)(3,4)$	$0$
$2$	$2$	$(1,5)(2,7)(3,8)(4,6)$	$0$
$2$	$2$	$(1,3)(2,4)(5,8)(6,7)$	$0$
$2$	$4$	$(1,7,2,5)(3,6,4,8)$	$0$
$2$	$4$	$(1,4,2,3)(5,8,7,6)$	$0$
$2$	$4$	$(1,6,2,8)(3,7,4,5)$	$0$
$2$	$4$	$(1,8,2,6)(3,7,4,5)$	$0$
$2$	$4$	$(1,7,2,5)(3,8,4,6)$	$0$
$2$	$4$	$(1,4,2,3)(5,6,7,8)$	$0$

The blue line marks the conjugacy class containing complex conjugation.